CASE III. To reduce integers or decimals to equivalent decimals of higher denominations. DIVIDE by the number of parts in the next higher denomination; continuing the operation to as many higher denominations as may be necessary, the same as in reduction ascending of whole numbers. 3. Reduce 7 drams to the decimal of a pound avoird. Ans. 02734375 lb. 4. Reduce ·26d to the decimal of a pound. 5. Reduce 2.15 lb to the decimal of a cwt. 6. Reduce 24 yards to the decimal of a mile. 10. Reduce .21 pint to the decimal of a peck. NOTE. highest. Ans. '001083361. Ans. 019196.... cwt. When there are several numbers, to be reduced all to the decimal of the Set the given numbers directly under each other, for dividends, proceeding orderly from the lowest denomination to the highest. Opposite to each dividend, on the left hand, set such a number for a divisor as will bring it to the next higher name; drawing a perpendicular line between all the divisors and dividends. Begin at the uppermost, and perform all the divisions: only observing to set the quotient of each division, as decimal parts, on the right hand of the dividend next below it; so shall the last quotient be the decimal required. RULE OF THREE IN DECIMALS. PREPARE the terms, by reducing the vulgar fractions to decimals, and any compound numbers either to decimals of the higher denominations, or to integers of the lower, also the first and third terms to the same name: then multiply and divide as in whole numbers. Note. Any of the convenient examples in the single or double rule of three in integers, or vulgar fractions, may be taken as proper examples to the same rules in decimals.-The following example, which is the first in vulgar fractions, is wrought out here, to show the method. DUODECIMALS, or CROSS MULTIPLICATION, is a rule used by workmen and artificers, in computing the contents of their works. Dimensions are usually taken in feet, inches, and quarters; any parts smaller than these being neglected as of no consequence. And the same in multiplying them together, or computing the contents. The method is as follows: SET down the two dimensions to be multiplied together, one under the other, so that feet may stand under feet, inches under inches, and so on. Multiply each term in the multiplicand, beginning at the lowest, by the feet in the multiplier, and set the result of each directly under its corresponding term, observing to carry 1 for every 12, from the inches to the feet. In like manner, multiply all the multiplicand by the inches and parts of the multiplier, and set the result of each term one place removed to the right-hand of those in the multiplicand; omitting, however, what is below parts of inches, only carrying to these the proper numbers of units from the lowest denomination. Or, instead of multiplying by the inches, take such parts of the multiplicand as these are of a foot. Then add the two lines together, after the manner of compound addition, carrying 1 to the feet for every 12 inches, when these come to so many. Note. The denomination which occupies the place of inches in these products, means not square inches, but rectangles of an inch broad and a foot long. Thus, the answer to the first example is 29 sq. feet, 4 sq. inches; to the second 66 sq. feet, 54 sq. inches. If the resulting product be one of three dimensions, length, breadth, and thickness, then the first denomination to the right of the feet must be multiplied by 144, the second by 12, and these products added to the figure in the third place, will give cubic inches *. *Though it is the practice to neglect all the smaller dimensions than inches or half inches both in actual measuring among artificers, and of course in the computations which are made from such surveys; yet in theory, all the subordinate dimensions are reckoned in a descending scale of twelves, as in our common numbers, we employ a descending scale of tens; and in all cases where the theoretical result is required, the process must be continued in the same way. Instead of the descending denominations below the units, tenths, hundredths, thousandths, &c. the terms, parts or primes, seconds, thirds, &c. descending successively below the inch, are employed. We are obliged, however, to keep the denominations in separate columns, or separated by a blank space, or by dots (,,) in our calculations in Duodecimals, instead of placing the numbers in each successive denomination in juxta-position as in our common notation. But if there were distinct and concentrated symbols employed to designate the numbers 10 and 11, (as Þor and II or π, as is done by some writers on the Theory of Numbers,) then we might dispense with the extra spaces, columns, or dots, and write the results continuously. All the advantages of the decimal notation in point of simplicity of writing would thus be gained for the duodecimal: and it is quite obvious that the same method is applicable to any other scale of numbers and its corresponding notation. The great barrier, however, to any change, except in the particular instance of feet and inches, is the terminology, or names of numbers, which could not possibly co-exist with a change of scale. The names are of an origin so decidedly and obviously decimal, that it requires some degree of fixed attention to ascertain how many dozens there are in any number specified decimally. All our language and all our ideas of number flow in terms of the decimal scale; and hence, however desirable it might appear in the eyes of some of the most enlightened mathematicians to adopt the duodecimal or dozen scale, the inveterate adherence which every people feels to its old language, and the consequent (in this case) daily practice, forbids even the most distant hope of ever realizing such a project. The intelligent teacher, however, may avail himself of the principle employed in duodecimals to explain to his more intelligent pupils the nature of numerical scales in general. Such pupils are now arrived at a stage in their arithmetical studies which renders such knowledge essential. A single specimen of the process of complete duodecimal multiplication expressed in the common and contracted notations is here subjoined, which it is hoped will give a clear idea of the views expressed above. The want of higher classes of twelves prevents our proceeding to the left without encountering the decimal notation for 10, 100, feet. 65 INVOLUTION. INVOLUTION is the raising of Powers from any given number, as a root. A Power is a quantity produced by multiplying any given number, called the Root, a certain number of times continually by itself. Thus, and in this manner may be calculated the following table of the first nine powers of the first 9 numbers. TABLE OF THE FIRST NINE POWERS OF THE FIRST NINE NUMBERS. The Index or Exponent of a power, is the number denoting the height or degree of that power; and it is 1 more than the number of multiplications used in producing the same. Thus 1 is the index or exponent of the 1st power or root, 2 of the 2nd power or square, 3 of the 3rd power or cube, 4 of the 4th power, or biquadrate, and so on. Powers, that are to be raised, are usually denoted by placing the index above the root or first power. So 22 4 is the 2d power of 2. 24 16 is the 4th power of 2. 540485030560000 is the 4th power of 540. When two or more powers are multiplied together, their product is that power whose index is the sum of the exponents of the factors or powers multiplied. Or the multiplication of the powers answers to the addition of the indices. Thus, in the following powers of 2, 2d 3d 4th 5th 6th 7th 8th 9th 10th 1st EVOLUTION, or the reverse of Involution, is the extracting or finding the roots of any given powers. The root of any number, or power, is such a number, as being multiplied into itself a certain number of times, will produce that power. Thus, 2 is the square root, or 2d root of 4, because 2? = 2 × 2 = 4; and 3 is the cube root or 3d root of 27, because 33 = 3 × 3 × 3 = 27. Any power of a given number or root may be found exactly, namely, by multiplying the number continually into itself. But there are many numbers of which a proposed root can never be exactly found; those numbers being themselves incapable of being produced by the involution (to the corresponding power) of any root composed of a finite number of integer or decimal places. Yet, by means of decimals, we may approximate or approach towards the root, to any assigned degree of exactness. Those roots which only approximate are called Surd Roots; but those which can be found quite exactly, are called Rational Roots. Thus, the square root of 3 is a surd root; but the square root of 4 is a rational root, being equal to 2: also the cube root of 8 is rational, being equal to 2; but the cube root of 9 is surd or irrational. Roots are sometimes denoted by writing the character ✓ before the power, with the index of the root against it. Thus, the 3d root of 20 is expressed by 20; and the square root or 2d root of it is √ 20, the index 2 being always omitted, when only the square root is designed. 12 is 3/45 When the power is expressed by several numbers, with the sign + or between them, a line is drawn from the top of the sign over all the parts of it: thus the third root of 45 12, or thus, / (45 12), enclosing the numbers in parentheses, which is, usually, the best way to express it. But all roots are now often designed like powers, with fractional indices; thus the square root of 8 is 8, the cube root of 25 is 25, and the 4th root of - 18 is (45. 45 18). TO EXTRACT THE SQUARE ROOT. * DIVIDE the given number into periods of two figures each, by setting a point over the place of units, another over the place of hundreds, and so on, * The reason for separating the figures of the dividend into periods or portions of two places each, is, that the square of any single figure never consists of more than two places; the square of a number of two figures, of not more than four places, and so on. So that there will be as many figures in the root as the given number contains periods so divided or parted off. And the reason of the several steps in the operation appears from the algebraic form of the square of any number of terms, whether two or three or more. Thus (a+b)2=a2 + 2ab + b2=,a2 + (2a + b) b, the square of two terms; where it appears that a is the first term of the root, and b the second term; also a the first divisor, and the new divisor |